Path Connected Space

Definition

A Topological Space $(X,\mathcal{T})$ is path connected if every pair of points of $X$ can be joined by a path $f:[0,1]\to X$ in $X$ where a path is a continuous map such that $f(0)=x$ and $f(1)=y$. $\forall x,y\in X,\exists f:[0,1]\to X\text{s.t.}f\in C^0,f(0)=x,f(1)=y$

Facts

Every path connected space is a Connected Space.

Every Convex Set in ${\mathbb{R}}^n$ is path connected.

Every open connected subset in ${\mathbb{R}}^n$ is path connected.

Consider a collection of path connected subspaces $\{U_i{\}}_{i\in I}\text{s.t.}{U}_i\subset X$ of a Topological Space $(X,\mathcal{T})$. If $\bigcap _{i\in I}{U}_i\ne \varnothing$, then $\bigcup _{i\in I}{U}_i$ is path connected.